D28:5S3 - GroupNames

G = D₂₈⋊₅S₃ order 336 = 2⁴·3·7

The semidirect product of D₂₈ and S₃ acting through Inn(D₂₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₈⋊₅S₃, D₁₄.₁D₆, D₆.₅D₁₄, C₂₈.₂₆D₆, Dic₄₂⋊₆C₂, C₁₂.₁₃D₁₄, C₄₂.₂C₂³, C₈₄.₁₂C₂², Dic₃.₁₀D₁₄, Dic₂₁.₂C₂², (C₄×S₃)⋊₁D₇, C₄.₅(S₃×D₇), (S₃×C₂₈)⋊₁C₂, (C₃×D₂₈)⋊₂C₂, C₃⋊₃(C₄○D₂₈), C₂₁⋊₁(C₄○D₄), C₂₁⋊D₄⋊₁C₂, C₇⋊₂(D₄⋊₂S₃), (Dic₃×D₇)⋊₁C₂, C₆.₂(C₂²×D₇), C₁₄.₂(C₂²×S₃), (C₆×D₇).₁C₂², (S₃×C₁₄).₅C₂², (C₇×Dic₃).₇C₂², C₂.₆(C₂×S₃×D₇), SmallGroup(336,138)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₂ — D₂₈⋊₅S₃

Chief series

C₁ — C₇ — C₂₁ — C₄₂ — C₆×D₇ — Dic₃×D₇ — D₂₈⋊₅S₃

Lower central

C₂₁ — C₄₂ — D₂₈⋊₅S₃

Upper central

C₁ — C₂ — C₄

Generators and relations for D₂₈⋊₅S₃
G = < a,b,c,d | a²⁸=b²=c³=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, dbd=a¹⁴b, dcd=c^-1 >

Subgroups: 420 in 80 conjugacy classes, 32 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₇, C₂×C₄, D₄, Q₈, Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, D₇, C₁₄, C₁₄, C₄○D₄, C₂₁, Dic₆, C₄×S₃, C₂×Dic₃, C₃⋊D₄, C₃×D₄, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, S₃×C₇, C₃×D₇, C₄₂, D₄⋊₂S₃, Dic₁₄, C₄×D₇, D₂₈, C₇⋊D₄, C₂×C₂₈, C₇×Dic₃, Dic₂₁, C₈₄, C₆×D₇, S₃×C₁₄, C₄○D₂₈, Dic₃×D₇, C₂₁⋊D₄, C₃×D₂₈, S₃×C₂₈, Dic₄₂, D₂₈⋊₅S₃
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, D₇, C₄○D₄, C₂²×S₃, D₁₄, D₄⋊₂S₃, C₂²×D₇, S₃×D₇, C₄○D₂₈, C₂×S₃×D₇, D₂₈⋊₅S₃

Smallest permutation representation of D₂₈⋊₅S₃
►On 168 points

Generators in S₁₆₈

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)

(1 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(106 112)(107 111)(108 110)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(132 140)(133 139)(134 138)(135 137)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 168)(159 167)(160 166)(161 165)(162 164)

(1 81 106)(2 82 107)(3 83 108)(4 84 109)(5 57 110)(6 58 111)(7 59 112)(8 60 85)(9 61 86)(10 62 87)(11 63 88)(12 64 89)(13 65 90)(14 66 91)(15 67 92)(16 68 93)(17 69 94)(18 70 95)(19 71 96)(20 72 97)(21 73 98)(22 74 99)(23 75 100)(24 76 101)(25 77 102)(26 78 103)(27 79 104)(28 80 105)(29 165 138)(30 166 139)(31 167 140)(32 168 113)(33 141 114)(34 142 115)(35 143 116)(36 144 117)(37 145 118)(38 146 119)(39 147 120)(40 148 121)(41 149 122)(42 150 123)(43 151 124)(44 152 125)(45 153 126)(46 154 127)(47 155 128)(48 156 129)(49 157 130)(50 158 131)(51 159 132)(52 160 133)(53 161 134)(54 162 135)(55 163 136)(56 164 137)

(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)

G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(132,140)(133,139)(134,138)(135,137)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,168)(159,167)(160,166)(161,165)(162,164), (1,81,106)(2,82,107)(3,83,108)(4,84,109)(5,57,110)(6,58,111)(7,59,112)(8,60,85)(9,61,86)(10,62,87)(11,63,88)(12,64,89)(13,65,90)(14,66,91)(15,67,92)(16,68,93)(17,69,94)(18,70,95)(19,71,96)(20,72,97)(21,73,98)(22,74,99)(23,75,100)(24,76,101)(25,77,102)(26,78,103)(27,79,104)(28,80,105)(29,165,138)(30,166,139)(31,167,140)(32,168,113)(33,141,114)(34,142,115)(35,143,116)(36,144,117)(37,145,118)(38,146,119)(39,147,120)(40,148,121)(41,149,122)(42,150,123)(43,151,124)(44,152,125)(45,153,126)(46,154,127)(47,155,128)(48,156,129)(49,157,130)(50,158,131)(51,159,132)(52,160,133)(53,161,134)(54,162,135)(55,163,136)(56,164,137), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)(132,140)(133,139)(134,138)(135,137)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)(158,168)(159,167)(160,166)(161,165)(162,164), (1,81,106)(2,82,107)(3,83,108)(4,84,109)(5,57,110)(6,58,111)(7,59,112)(8,60,85)(9,61,86)(10,62,87)(11,63,88)(12,64,89)(13,65,90)(14,66,91)(15,67,92)(16,68,93)(17,69,94)(18,70,95)(19,71,96)(20,72,97)(21,73,98)(22,74,99)(23,75,100)(24,76,101)(25,77,102)(26,78,103)(27,79,104)(28,80,105)(29,165,138)(30,166,139)(31,167,140)(32,168,113)(33,141,114)(34,142,115)(35,143,116)(36,144,117)(37,145,118)(38,146,119)(39,147,120)(40,148,121)(41,149,122)(42,150,123)(43,151,124)(44,152,125)(45,153,126)(46,154,127)(47,155,128)(48,156,129)(49,157,130)(50,158,131)(51,159,132)(52,160,133)(53,161,134)(54,162,135)(55,163,136)(56,164,137), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(106,112),(107,111),(108,110),(113,131),(114,130),(115,129),(116,128),(117,127),(118,126),(119,125),(120,124),(121,123),(132,140),(133,139),(134,138),(135,137),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150),(158,168),(159,167),(160,166),(161,165),(162,164)], [(1,81,106),(2,82,107),(3,83,108),(4,84,109),(5,57,110),(6,58,111),(7,59,112),(8,60,85),(9,61,86),(10,62,87),(11,63,88),(12,64,89),(13,65,90),(14,66,91),(15,67,92),(16,68,93),(17,69,94),(18,70,95),(19,71,96),(20,72,97),(21,73,98),(22,74,99),(23,75,100),(24,76,101),(25,77,102),(26,78,103),(27,79,104),(28,80,105),(29,165,138),(30,166,139),(31,167,140),(32,168,113),(33,141,114),(34,142,115),(35,143,116),(36,144,117),(37,145,118),(38,146,119),(39,147,120),(40,148,121),(41,149,122),(42,150,123),(43,151,124),(44,152,125),(45,153,126),(46,154,127),(47,155,128),(48,156,129),(49,157,130),(50,158,131),(51,159,132),(52,160,133),(53,161,134),(54,162,135),(55,163,136),(56,164,137)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159)]])

51 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	6A	6B	6C	7A	7B	7C	12	14A	14B	14C	14D	···	14I	21A	21B	21C	28A	···	28F	28G	···	28L	42A	42B	42C	84A	···	84F
order	1	2	2	2	2	3	4	4	4	4	4	6	6	6	7	7	7	12	14	14	14	14	···	14	21	21	21	28	···	28	28	···	28	42	42	42	84	···	84
size	1	1	6	14	14	2	2	3	3	42	42	2	28	28	2	2	2	4	2	2	2	6	···	6	4	4	4	2	···	2	6	···	6	4	4	4	4	···	4

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+	+		-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	C₄○D₂₈	D₄⋊₂S₃	S₃×D₇	C₂×S₃×D₇	D₂₈⋊₅S₃
kernel	D₂₈⋊₅S₃	Dic₃×D₇	C₂₁⋊D₄	C₃×D₂₈	S₃×C₂₈	Dic₄₂	D₂₈	C₂₈	D₁₄	C₄×S₃	C₂₁	Dic₃	C₁₂	D₆	C₃	C₇	C₄	C₂	C₁
# reps	1	2	2	1	1	1	1	1	2	3	2	3	3	3	12	1	3	3	6

Matrix representation of D₂₈⋊₅S₃ ►in GL₆(𝔽₃₃₇)

178	196	0	0	0	0
12	159	0	0	0	0
0	0	33	227	0	0
0	0	110	1	0	0
0	0	0	0	336	0
0	0	0	0	0	336

336	0	0	0	0	0
160	1	0	0	0	0
0	0	304	110	0	0
0	0	33	33	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	336
0	0	0	0	1	336

58	26	0	0	0	0
91	279	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	329	214
0	0	0	0	206	8

G:=sub<GL(6,GF(337))| [178,12,0,0,0,0,196,159,0,0,0,0,0,0,33,110,0,0,0,0,227,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[336,160,0,0,0,0,0,1,0,0,0,0,0,0,304,33,0,0,0,0,110,33,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,336,336],[58,91,0,0,0,0,26,279,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,329,206,0,0,0,0,214,8] >;

D₂₈⋊₅S₃ in GAP, Magma, Sage, TeX

D_{28}\rtimes_5S_3

% in TeX

G:=Group("D28:5S3");

// GroupNames label

G:=SmallGroup(336,138);

// by ID

G=gap.SmallGroup(336,138);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,50,490,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^14*b,d*c*d=c^-1>;

// generators/relations

G = D28⋊5S3 order 336 = 24·3·7

The semidirect product of D28 and S3 acting through Inn(D28)

G = D₂₈⋊₅S₃ order 336 = 2⁴·3·7

The semidirect product of D₂₈ and S₃ acting through Inn(D₂₈)